import random
import os
random.seed()
from math import *


def findNs(a, b):
  p = a*b
  i = int(floor((log(p)/2)-1))
  j = int(floor(log(p)*2))
  limit = ceil((p**0.5) / (floor(log(p)/2)-1))
  limit = ceil(limit**(1/(3**0.5)))
  s = ''
  ns = ''
  n = 0
  k = 0
  while i <= limit:
    j = int((int(floor(log(p))*2)**2)/2)
    while j >= 3:
      ns = str(p)+str(i)+(str(p)[0:-1])+str(j)
      n = int(ns)
      valA = (n%(p-1))%a
      valB = (n%(p-1))%b
      if valA == 0 or valA == 1 or valB == 0 or valB == 1:
        _ = input("found valA or valB that equals either 1 or 0")
      print(f"ns%a: {ns}, val: {valA},  ij: {i}, {j}")
      print(f"ns%b: {ns}, val: {valB},  ij: {i}, {j}")
      if (n%(p-1))%a == 0 or (n%(p-1))%a == 1 or (n%(p-1))%b == 0 or (n%(p-1))%a == 1:
        print(f"found ns value that works!,  total iterations: {k}, ratio over a: {k/a},  ij: {i}, {j}")
        print(f"limit: {limit}")
        print(f"p: {p}, k: {k}")
        return i, j, n, limit
      else:
        j = j - 1
        k = k + 1
    i = i + 1

  #else
  print("no exact match found. Returing None.")
  return None